Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2018 Winter 4th Sem Old 140001 Mathematics Iv Previous Question Paper
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (OLD) EXAMINATION ? WINTER 2018
Subject Code:140001 Date: 22/11/2018
Subject Name: Mathematics-IV
Time: 02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) (i) Define harmonic function and show that ?? (?? , ?? ) = ?? 2
? ?? 2
is
harmonic.
03
(ii) Show that ?????? ?? ?0
?? ? ?? does not exist.
04
(b) Apply Gauss-Seidel method to solve the equations
20?? + ?? ? 2?? = 17, 3?? + 20?? ? ?? = ?18, 2?? ? 3?? + 20?? = 25.
07
Q.2 (a) (i) State Trapezoidal rule, Simpson?s 1/3
rd
rule and Simpson 3/8
th
rule. 03
(ii) Find the positive root of ?? 3
? ?? ? 11 = 0 correct to the three decimal
places by Bisection method.
04
(b) Obtain approximate value of ?? at ?? = 0.2 for the differential equation
????
????
= 2?? + 3?? ?? , ?? (0) = 0 , using Taylor?s method. Also compare the
obtained numerical solution with the exact solution.
07
OR
(b)
Using Runge-Kutta method of fourth order, solve
????
????
=
?? 2
??? 2
?? 2
+?? 2
with
?? (0) = 1 at ?? = 0.2.
07
Q.3 (a)
(i) Evaluate ? ?? ??? 2 1
0
???? using Gauss quadrature formula of three points.
03
(ii) Using usual notations, show that ?= ?? ?= ?? ?= ?? ?? 1/2
and (?? 1/2
+
?? ?1/2
) (1 + ?)
1/2
= 2 + ?.
04
(b) Using Lagrange?s interpolation formula, find the value of ?? when ?? =
10, if the following values of ?? and ?? are given:
?? 5 6 9 11
?? 12 13 14 16
07
OR
Q.3 (a) (i) Define entire function and show that ?? (?? ) = ?? ?? is entire function. 03
(ii) Find harmonic conjugate of ?? (?? , ?? ) = 2?? ? ?? 3
+ 3?? ?? 2
. 04
(b) Using Newton forward interpolation formula, find the value of ?? (1.6) if
the following values of ?? and ?? (?? ) are given:
?? 1 1.4 1.8 2.2
?? (?? ) 3.49 4.82 5.96 6.5
07
Q.4 (a)
(i) Evaluate ?
?? ??? ?? +1 ?? ???? , where ?? is a circle |?? | = 1/2.
03
(ii) Determine and sketch the image of |?? | = 1 under the
transformation?? = ?? + ?? .
04
(b) Find Maclaurin?s series expansion of the function ?? (?? ) = ?????? 2
?? . 07
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (OLD) EXAMINATION ? WINTER 2018
Subject Code:140001 Date: 22/11/2018
Subject Name: Mathematics-IV
Time: 02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) (i) Define harmonic function and show that ?? (?? , ?? ) = ?? 2
? ?? 2
is
harmonic.
03
(ii) Show that ?????? ?? ?0
?? ? ?? does not exist.
04
(b) Apply Gauss-Seidel method to solve the equations
20?? + ?? ? 2?? = 17, 3?? + 20?? ? ?? = ?18, 2?? ? 3?? + 20?? = 25.
07
Q.2 (a) (i) State Trapezoidal rule, Simpson?s 1/3
rd
rule and Simpson 3/8
th
rule. 03
(ii) Find the positive root of ?? 3
? ?? ? 11 = 0 correct to the three decimal
places by Bisection method.
04
(b) Obtain approximate value of ?? at ?? = 0.2 for the differential equation
????
????
= 2?? + 3?? ?? , ?? (0) = 0 , using Taylor?s method. Also compare the
obtained numerical solution with the exact solution.
07
OR
(b)
Using Runge-Kutta method of fourth order, solve
????
????
=
?? 2
??? 2
?? 2
+?? 2
with
?? (0) = 1 at ?? = 0.2.
07
Q.3 (a)
(i) Evaluate ? ?? ??? 2 1
0
???? using Gauss quadrature formula of three points.
03
(ii) Using usual notations, show that ?= ?? ?= ?? ?= ?? ?? 1/2
and (?? 1/2
+
?? ?1/2
) (1 + ?)
1/2
= 2 + ?.
04
(b) Using Lagrange?s interpolation formula, find the value of ?? when ?? =
10, if the following values of ?? and ?? are given:
?? 5 6 9 11
?? 12 13 14 16
07
OR
Q.3 (a) (i) Define entire function and show that ?? (?? ) = ?? ?? is entire function. 03
(ii) Find harmonic conjugate of ?? (?? , ?? ) = 2?? ? ?? 3
+ 3?? ?? 2
. 04
(b) Using Newton forward interpolation formula, find the value of ?? (1.6) if
the following values of ?? and ?? (?? ) are given:
?? 1 1.4 1.8 2.2
?? (?? ) 3.49 4.82 5.96 6.5
07
Q.4 (a)
(i) Evaluate ?
?? ??? ?? +1 ?? ???? , where ?? is a circle |?? | = 1/2.
03
(ii) Determine and sketch the image of |?? | = 1 under the
transformation?? = ?? + ?? .
04
(b) Find Maclaurin?s series expansion of the function ?? (?? ) = ?????? 2
?? . 07
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2
OR
Q.4 (a)
(i) Evaluate ?
?? ?? ?? +1 ?? ???? , where ?? is a circle |?? | = 2.
03
(ii) Find bilinear transformation that maps the points -1, 0, 1 onto ?1, -i,1
respectively.
04
(b)
Expand ?? (?? ) =
1
(?? +1)(?? +3)
in Laurent?s series valid for |?? | < 1 and 1 <
|?? | < 3.
07
Q.5 (a)
(i) Find an upper bound for the absolute value of the integral ? ?? ?? ???? ,
??
where C is the line segment joining the points (0,0) and (1, 2?2).
03
(ii) Determine the poles of the function ?? (?? ) =
1
(?? 2
+1)
2
and the residue at
each poles.
04
(b)
Evaluate ?
????
(?? 2
+1) (?? 2
+2?? +2)
?
? ?
.
07
OR
Q.5 (a) (i) State (i) Rouche?s Thorem (ii) Liouville?s Theorem 03
(ii) Evaluate ?
?? 3?? (?? ????? 2)
4
???? ?? , where C is the square with vertices at ?1, ??? .
04
(b)
Evaluate ?
????
(2+???????? )
2
2?? 0
.
07
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This post was last modified on 20 February 2020